Author Topic: Turning inexact solutions into exact solutions  (Read 1400 times)

Yasmine Hemmati

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Turning inexact solutions into exact solutions
« on: September 26, 2018, 03:15:55 PM »
How do integrating factors turn an inexact solution into an exact solution.
« Last Edit: September 26, 2018, 04:30:57 PM by Victor Ivrii »

Victor Ivrii

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Re: Turning inexact solutions into exact solutions
« Reply #1 on: September 26, 2018, 04:34:27 PM »
The question does not make any sense because there are no exact or non-exact solutions, but there are exact or non-exact equations.

The original equation and this equation multiplied by an integrating factor have the same solutions, but the first one is not exact, and the second is exact.

Wei Cui

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Re: Turning inexact solutions into exact solutions
« Reply #2 on: September 27, 2018, 01:43:07 AM »
If you ask how to turn the inexact equation into exact equation, then there are three cases:

Try 1: check $\frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} =$ $f(x)$ function of $x$ only,
 
then let $\frac{u^{'}}{u} = f(x)$
           
           $\frac{du}{u} = f(x)dx$

            $u = e^{\int f(x)dx}$

Try 2: check $\frac{\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}}{M} = g(y)$ function of $y$ only, and then same as the first one you let $\frac{u^{'}}{u} = g(y)$,
         
          $u = e^{\int g(y)dy}$

Try 3: if $\frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{Mx-Ny} = z(x, y)$, then
         
          $u = u(x, y)$ and $u = e^{\int z(x,y)}$

In each of these cases,  $u$ is the integrating factor, when you solve $u$ and you multiply the equation both sides with $u$ then you will turn the inexact equation into an exact equation.
« Last Edit: September 27, 2018, 01:44:39 AM by Wei Cui »

Victor Ivrii

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Re: Turning inexact solutions into exact solutions
« Reply #3 on: September 27, 2018, 02:39:44 AM »
There are three cases for which on the lectures and in the textbook the recipe was provided. There are some other cases. And for some equations the simple guess will work.